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The Immediate Exchange model: an analytical investigation

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  • Guy Katriel

Abstract

We study the Immediate Exchange model, recently introduced by Heinsalu and Patriarca [Eur. Phys. J. B 87: 170 (2014)], who showed by simulations that the wealth distribution in this model converges to a Gamma distribution with shape parameter $2$. Here we justify this conclusion analytically, in the infinite-population limit. An infinite-population version of the model is derived, describing the evolution of the wealth distribution in terms of iterations of a nonlinear operator on the space of probability densities. It is proved that the Gamma distributions with shape parameter $2$ are fixed points of this operator, and that, starting with an arbitrary wealth distribution, the process converges to one of these fixed points. We also discuss the mixed model introduced in the same paper, in which exchanges are either bidirectional or unidirectional with fixed probability. We prove that, although, as found by Heinsalu and Patriarca, the equilibrium distribution can be closely fit by Gamma distributions, the equilibrium distribution for this model is {\it{not}} a Gamma distribution.

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  • Guy Katriel, 2014. "The Immediate Exchange model: an analytical investigation," Papers 1409.6646, arXiv.org.
    • Handle: RePEc:arx:papers:1409.6646
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    References listed on IDEAS

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    1. Adrian Dragulescu & Victor M. Yakovenko, 2000. "Statistical mechanics of money," Papers cond-mat/0001432, arXiv.org, revised Aug 2000.
    2. Mehdi Lallouache & Aymen Jedidi & Anirban Chakraborti, 2010. "Wealth distribution: To be or not to be a Gamma?," Papers 1004.5109, arXiv.org, revised May 2010.
    3. Guy Katriel, 2014. "Directed Random Market: the equilibrium distribution," Papers 1404.4068, arXiv.org.
    4. M. Patriarca & E. Heinsalu & A. Chakraborti, 2010. "Basic kinetic wealth-exchange models: common features and open problems," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 73(1), pages 145-153, January.
    5. Els Heinsalu & Marco Patriarca, 2014. "Kinetic models of immediate exchange," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 87(8), pages 1-10, August.
    6. Victor M. Yakovenko & J. Barkley Rosser, 2009. "Colloquium: Statistical mechanics of money, wealth, and income," Papers 0905.1518, arXiv.org, revised Dec 2009.
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    Cited by:

    1. Redig, Frank & Sau, Federico, 2017. "Generalized immediate exchange models and their symmetries," Stochastic Processes and their Applications, Elsevier, vol. 127(10), pages 3251-3267.
    2. Max Greenberg & H. Oliver Gao, 2024. "Twenty-five years of random asset exchange modeling," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 97(6), pages 1-27, June.

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